Example 3
Example 3
This can be written as
u'=Au
where
u={x1,x2,x3,x4}
A={{2,-1,2,0},{0,3,-1,0},{0,1,1,0},{0,1,-3,5}}
u0={-1,1,1,0}
In[79]:=
a={{2,-1,2,0},{0,3,-1,0},{0,1,1,0},{0,1,-3,5}};
MatrixForm[a]
u0={-1,1,1,0}
Out[79]=
2 -1 2 0 0 3 -1 0 0 1 1 0 0 1 -3 5
Out[80]=
{-1, 1, 1, 0}
Let's use Mathematica's tools.
In[81]:=
{s,j}=JordanDecomposition[a];
MatrixForm[j]
MatrixForm[s]
Out[81]=
2 1 0 0 0 2 1 0 0 0 2 0 0 0 0 5
Out[82]=
1 0 0 0
0 1 2 0
0 1 1 0
2 5
- -
0 3 9 1
In[83]:=
expta=MatrixExp[t a];
MatrixForm[expta]
Out[83]=
2 t
E
2 t 2
2 t E t
-(E t) + -------
2
2 t 2
2 t E t
2 E t - -------
2
0
0
2 t 2 t 2
2 t -(E t) + E t
2 E + -------------------
t
2 t 2 t 2
2 t 2 E t - E t
-2 E + ------------------
t
0
0
2 t 2 t 2
2 t -(E t) + E t
E + -------------------
t
2 t 2 t 2
2 t 2 E t - E t
-E + ------------------
t
0
0
2 t 5 t 2 t 2 t 2
5 E E 2 (-(E t) + E t )
------ + ---- + -----------------------
9 9 3 t
2 t 5 t 2 t 2 t 2
-5 E 7 E 2 (2 E t - E t )
------- - ------ + ----------------------
9 9 3 t
5 t
E
In[84]:=
sol=expta.u0
Out[84]=
2 t 2 t 2 2 t 2 t 2
2 t 2 t 2 E t - E t -(E t) + E t
{-E + E t, ------------------ + -------------------,
t t
2 t 2 t 2 2 t 2 t 2
2 E t - E t -(E t) + E t
------------------ + -------------------,
t t
5 t 2 t 2 t 2
-2 E 2 (2 E t - E t )
------- + ---------------------- +
3 3 t
2 t 2 t 2
2 (-(E t) + E t )
-----------------------}
3 t
Let's check.
In[85]:=
Simplify[D[sol,{t,1}]-a.sol]
Out[85]=
{0, 0, 0, 0}
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